AMC 8 2011 Problem 3
AMC8 2011 年第 3 题
四边形与多边形★★☆☆☆
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
把下图所示的有 8 块黑色和 17 块白色方形瓷砖组成的正方形图案，通过围绕着这个大正方形的外周贴一圈黑色的瓷砖而进行拓展。那么拓展后的图形中，黑色瓷砖和白色瓷砖的个数的比值是多少？
- A.8:17
- B.25:49
- C.36:25
- D.32:17
- E.36:17
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
One way of approaching this is drawing the next circle of boxes around the current square. We can now count the number of black and white tiles; 32 black tiles and 17 white tiles. This means the answer is .
If we did not want to draw a diagram though that is probably simpler in this case, we can imagine the last border of black tiles. We see that each side length (if the side length of a square is 1 ) increases by 2 each time we go out one layer. Therefore, we know that the next border will have a side length of 7. That border has black squares in it. The other black border has squares in it, so there are 32 black squares in all. The other ones must all be white, so there are 49-32=17 white squares. Thus, we get our answer of . Notice that in total, once we add the border there will be squares total. So far as there is only one option that sums to 49 and we know that there are an integer number of boxes, it has to be .
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